In this infinite tree, degree of each vertex is equal to 3. A real number $ \lambda$ is given. We want to assign a real number to each node in such a way that for each node sum of numbers assigned to its neighbors is equal to $ \lambda$ times of the number assigned to this node. Find all $ \lambda$ for which this is possible.

Put $ M\equal{}\begin{pmatrix}p&q&r\\ r&p&q\\q&r&p\end{pmatrix}$ where $ p,q,r>0$ and $ p\plus{}q\plus{}r\equal{}1$. Prove that $ \lim_{n\rightarrow \infty}M^n\equal{}\begin{bmatrix}\frac13&\frac13&\frac13\\ \frac13&\frac13&\frac13\\\frac13&\frac13&\frac13\end{bmatrix}$

Let $H{}$ be the orthocenter of the triangle $ABC{}$ and $X{}$ be the midpoint of the side $BC.$ The perpendicular at $H{}$ to $HX{}$ intersects the sides $(AB)$ and $(AC)$ at $Y{}$ and $Z{}$ respectively. Let $O{}$ be the circumcenter of $ABC{}$ and $O'$ be the circumcenter of $BHC.$ [list=a] [*]Prove that $HY=HZ.$ [*]Prove that $\overrightarrow{AY}+\overrightarrow{AZ}=2\overrightarrow{OO'}.$ [/list]

Let a quardilateral $ABCD$ with $AB=AD$ and $\widehat B=\widehat D=90$. At $CD$ we take point $E$ and at $BC$ we take point $Z$ such that $AE\bot DZ$. Prove that $AZ\bot BE$

XOY is angle in the plane.A,B are variable point on OX,OY such that 1/OA+1/OB=1/K (k is constant).draw two circles with diameter OA and OB.prove that common external tangent to these circles is tangent to the constant circle( ditermine the radius and the locus of its center).

We define an operation $\oplus$ on the set $\{0, 1\}$ by \[ 0 \oplus 0 = 0 \,, 0 \oplus 1 = 1 \,, 1 \oplus 0 = 1 \,, 1 \oplus 1 = 0 \,.\] For two natural numbers $a$ and $b$, which are written in base $2$ as $a = (a_1a_2 \ldots a_k)_2$ and $b = (b_1b_2 \ldots b_k)_2$ (possibly with leading 0's), we define $a \oplus b = c$ where $c$ written in base $2$ is $(c_1c_2 \ldots c_k)_2$ with $c_i = a_i \oplus b_i$, for $1 \le i \le k$. For example, we have $7 \oplus 3 = 4$ since $ 7 = (111)_2$ and $3 = (011)_2$. For a natural number $n$, let $f(n) = n \oplus \left[ n/2 \right]$, where $\left[ x \right]$ denotes the largest integer less than or equal to $x$. Prove that $f$ is a bijection on the set of natural numbers.

The National Foundation of Happiness (NFoH) wants to estimate the happiness of people of country. NFoH selected $n$ random persons, and on every morning asked from each of them whether she is happy or not. On any two distinct days, exactly half of the persons gave the same answer. Show that after $k$ days, there were at most $n-\frac{n}{k}$ persons whose “yes” answers equals their “no” answers.

Let $ABCD$ and $A'B'C'D'$ be two arbitrary squares in the plane that are oriented in the same direction. Prove that the quadrilateral formed by the midpoints of $AA',BB',CC',DD'$ is a square.

Suppose that the components of he vector $ \textbf{u}=(u_0,\ldots,u_n)$ are real functions defined on the closed interval $ [a,b]$ with the property that every nontrivial linear combination of them has at most $ n$ zeros in $ [a,b]$. Prove that if $ \sigma$ is an increasing function on $ [a,b]$ and the rank of the operator \[ A(f)= \int_{a}^b \textbf{u}(x)f(x)d\sigma(x), \;f \in C[a,b]\ ,\] is $ r \leq n$, then $ \sigma$ has exactly $ r$ points of increase. [i]E. Gesztelyi[/i]

The centre $O$ of the circumcircle of $\vartriangle ABC$ lies inside the triangle. Perpendiculars are drawn rom $O$ on the sides. When produced beyond the sides they meet the circumcircle at points $K, M$ and $P$. Prove that $\overrightarrow{OK} + \overrightarrow{OM} + \overrightarrow{OP} = \overrightarrow{OI}$, where $I$ is the centre of the inscribed circle of $\vartriangle ABC$. (V Galperin, Moscow)

Let $ p(x)\equal{}a_0 \plus{}a_1 x\plus{}...\plus{}a_n x^n$ be a polynomial with nonnegative real coefficients. Suppose that $ p(4)\equal{}2$ and $ p(16)\equal{}8$. Prove that $ p(8) \le 4$ and find all such $ p$ with $ p(8)\equal{}4$.

Let $ I$ be an ideal of the ring $\mathbb{Z}\left[x\right]$ of all polynomials with integer coefficients such that a) the elements of $ I$ do not have a common divisor of degree greater than $ 0$, and b) $ I$ contains of a polynomial with constant term $ 1$. Prove that $ I$ contains the polynomial $ 1 + x + x^2 + ... + x^{r-1}$ for some natural number $ r$. [i]Gy. Szekeres[/i]

Let $R$ be a commutative ring with $1$ such that the number of elements of $R$ is equal to $p^3$ where $p$ is a prime number. Prove that if the number of elements of $\text{zd}(R)$ be in the form of $p^n$ ($n \in \mathbb{N^*}$) where $\text{zd}(R) = \{a \in R \mid \exists 0 \neq b \in R, ab = 0\}$, then $R$ has exactly one maximal ideal.

Given vectors $\overline a,\overline b,\overline c\in\mathbb R^n$, show that $$(\lVert\overline a\rVert\langle\overline b,\overline c\rangle)^2+(\lVert\overline b\rVert\langle\overline a,\overline c\rangle)^2\le\lVert\overline a\rVert\lVert\overline b\rVert(\lVert\overline a\rVert\lVert\overline b\rVert+|\langle\overline a,\overline b\rangle|)\lVert\overline c\rVert^2$$where $\langle\overline x,\overline y\rangle$ denotes the scalar (inner) product of the vectors $\overline x$ and $\overline y$ and $\lVert\overline x\rVert^2=\langle\overline x,\overline x\rangle$.

Let $A$ be a $n\times n$ matrix with complex elements and let $A^\star$ be the classical adjoint of $A$. Prove that if there exists a positive integer $m$ such that $(A^\star)^m = 0_n$ then $(A^\star)^2 = 0_n$. [i]Marian Ionescu, Pitesti[/i]

The points $S(i, j)$ with integer Cartesian coordinates $0 < i \leq n, 0 < j \leq m, m \leq n$, form a lattice. Find the number of: [b](a)[/b] rectangles with vertices on the lattice and sides parallel to the coordinate axes; [b](b)[/b] squares with vertices on the lattice and sides parallel to the coordinate axes; [b](c)[/b] squares in total, with vertices on the lattice.

Let $a_{i}$ and $b_{i}$ ($i=1,2, \cdots, n$) be rational numbers such that for any real number $x$ there is: \[x^{2}+x+4=\sum_{i=1}^{n}(a_{i}x+b)^{2}\] Find the least possible value of $n$.

Polyhedron $ABCDEFG$ has six faces. Face $ABCD$ is a square with $AB=12;$ face $ABFG$ is a trapezoid with $\overline{AB}$ parallel to $\overline{GF},$ $BF=AG=8,$ and $GF=6;$ and face $CDE$ has $CE=DE=14.$ The other three faces are $ADEG, BCEF,$ and $EFG.$ The distance from $E$ to face $ABCD$ is 12. Given that $EG^2=p-q\sqrt{r},$ where $p, q,$ and $r$ are positive integers and $r$ is not divisible by the square of any prime, find $p+q+r.$

Consider a Riemannian metric on the vector space ${\Bbb{R}^n}$ which satisfies the property that for each two points ${a,b}$ there is a single distance minimising geodesic segment ${g(a,b)}$. Suppose that for all ${a \in \Bbb{R}^n}$, the Riemannian distance with respect to ${a}, {\rho_a : \Bbb{R}^n \rightarrow \Bbb{R}}$ is convex and differentiable outside of ${a}$. Prove that if for a point ${x \neq a,b}$ we have \[ \displaystyle \partial_i \rho_a(x)=-\partial_i \rho_b(x),\ i=1,\cdots, n\] then ${x}$ is a point on ${g(a,b)}$ and conversely. [i]Proposed by Lajos Tamássy and Dávid Kertész[/i]

A set $ E$ of points in the 3D space let $ L(E)$ denote the set of all those points which lie on lines composed of two distinct points of $ E.$ Let $ T$ denote the set of all vertices of a regular tetrahedron. Which points are in the set $ L(L(T))?$

Let $\rho:G\to GL(V)$ be a representation of a finite $p$-group $G$ over a field of characteristic $p$. Prove that if the restriction of the linear map $\sum_{g\in G} \rho(g)$ to a finite dimensional subspace $W$ of $V$ is injective, then the subspace spanned by the subspaces $\rho(g)W$ $(g\in G)$ is the direct sum of these subspaces.

Let $ P(X) \equal{} a_{n}X^{n} \plus{} a_{n \minus{} 1}X^{n \minus{} 1} \plus{} \ldots \plus{} a_{1}X \plus{} a_{0}$ be a real polynomial of degree $ n$. Suppose $ n$ is an even number and: a) $ a_{0} > 0$, $ a_{n} > 0$; b) $ a_{1}^{2} \plus{} a_{2}^{2} \plus{} \ldots \plus{} a_{n \minus{} 1}^{2}\leq\frac {4\min(a_{0}^{2} , a_{n}^{2})}{n \minus{} 1}$. Prove that $ P(x)\geq 0$ for all real values $ x$. [i]Laurentiu Panaitopol[/i]

Let $m,n\in\mathbb{Z_{+}}$ be such numbers that set $\{1,2,\ldots,n\}$ contains exactly $m$ different prime numbers. Prove that if we choose any $m+1$ different numbers from $\{1,2,\ldots,n\}$ then we can find number from $m+1$ choosen numbers, which divide product of other $m$ numbers.

For an integer $n\geq 3$, let $\theta=2\pi/n$. Evaluate the determinant of the $n\times n$ matrix $I+A$, where $I$ is the $n\times n$ identity matrix and $A=(a_{jk})$ has entries $a_{jk}=\cos(j\theta+k\theta)$ for all $j,k$.

Let $n\ge 3$ be an integer. Two players, Ana and Beto, play the following game. Ana tags the vertices of a regular $n$- gon with the numbers from $1$ to $n$, in any order she wants. Every vertex must be tagged with a different number. Then, we place a turkey in each of the $n$ vertices. These turkeys are trained for the following. If Beto whistles, each turkey moves to the adjacent vertex with greater tag. If Beto claps, each turkey moves to the adjacent vertex with lower tag. Beto wins if, after some number of whistles and claps, he gets to move all the turkeys to the same vertex. Ana wins if she can tag the vertices so that Beto can't do this. For each $n\ge 3$, determine which player has a winning strategy. [i]Proposed by Victor and Isaías de la Fuente[/i]